the fixed charge transportation problem: a strong ...very few papers on lagrangian decomposition and...
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Introduction and background The fixed charge transportation problem Concluding comments
The Fixed Charge Transportation Problem:A Strong Formulation Based On LagrangianDecomposition and Column Generation
Yixin Zhao, Torbjorn Larsson and Elina RonnbergDepartment of Mathematics, Linkoping University, Sweden
Column generation 2016
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
What is this talk about?
Strong lower bounding for the fixed charge transportation problem by
I Lagrangian decomposition:supply and demand side copies of the shipping variables
I Dual cutting plane method (column generation)
Why?
I Lagrangian decomposition can give strong formulations
I Strong formulations of interest in column generation
I Combination not utilised in many papers
I Preliminary work to study the strength of the formulation:theoretically and empirically
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
What is this talk about?
Strong lower bounding for the fixed charge transportation problem by
I Lagrangian decomposition:supply and demand side copies of the shipping variables
I Dual cutting plane method (column generation)
Why?
I Lagrangian decomposition can give strong formulations
I Strong formulations of interest in column generation
I Combination not utilised in many papers
I Preliminary work to study the strength of the formulation:theoretically and empirically
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Outline
Introduction and background
The fixed charge transportation problem
Concluding comments
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Lagrangian decomposition / Lagrangian relaxationConsider the problem
(P) min{cx s.t. Ax ≤ b, Cx ≤ d , x ∈ X} =min{cx s.t. Ay ≤ b, Cx ≤ d , x = y , x ∈ X , y ∈ Y },where Y is such that X ⊆ Y
Lagrangian decomposition
(LD) min{cx + u(x − y) s.t. Ay ≤ b, Cx ≤ d , x ∈ X , y ∈ Y } =min{(c + u)x s.t. Cx ≤ d , x ∈ X}+ min{−uy s.t. Ay ≤ b, y ∈ Y }
Lagrangian relaxation w.r.t. one of the constraint groups,for example Ax ≤ b
(LR) min{cx + v(Ax − b) s.t. Cx ≤ d , x ∈ X} =min{(c + vA)x s.t. Cx ≤ d , x ∈ X}+ vb,where v ≥ 0
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Lagrangian decomposition / Lagrangian relaxationConsider the problem
(P) min{cx s.t. Ax ≤ b, Cx ≤ d , x ∈ X} =min{cx s.t. Ay ≤ b, Cx ≤ d , x = y , x ∈ X , y ∈ Y },where Y is such that X ⊆ Y
Lagrangian decomposition
(LD) min{cx + u(x − y) s.t. Ay ≤ b, Cx ≤ d , x ∈ X , y ∈ Y } =min{(c + u)x s.t. Cx ≤ d , x ∈ X}+ min{−uy s.t. Ay ≤ b, y ∈ Y }
Lagrangian relaxation w.r.t. one of the constraint groups,for example Ax ≤ b
(LR) min{cx + v(Ax − b) s.t. Cx ≤ d , x ∈ X} =min{(c + vA)x s.t. Cx ≤ d , x ∈ X}+ vb,where v ≥ 0
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Lagrangian decomposition / Lagrangian relaxationConsider the problem
(P) min{cx s.t. Ax ≤ b, Cx ≤ d , x ∈ X} =min{cx s.t. Ay ≤ b, Cx ≤ d , x = y , x ∈ X , y ∈ Y },where Y is such that X ⊆ Y
Lagrangian decomposition
(LD) min{cx + u(x − y) s.t. Ay ≤ b, Cx ≤ d , x ∈ X , y ∈ Y } =min{(c + u)x s.t. Cx ≤ d , x ∈ X}+ min{−uy s.t. Ay ≤ b, y ∈ Y }
Lagrangian relaxation w.r.t. one of the constraint groups,for example Ax ≤ b
(LR) min{cx + v(Ax − b) s.t. Cx ≤ d , x ∈ X} =min{(c + vA)x s.t. Cx ≤ d , x ∈ X}+ vb,where v ≥ 0
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Strength of bounds
[Guignard and Kim(1987)]: The Lagrangian decomposition bound isas least as strong as the strongest of
I the Lagrangian relaxation bound when Ax ≤ b is relaxed
I the Lagrangian relaxation bound when Cx ≤ d is relaxed
and there is a chance that it is stronger!
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Related work
Very few papers on Lagrangian decomposition and column generation:
I [Pimentel et al.(2010)]:The multi-item capacitated lot sizing problem
− Branch-and-price implementations for two types of Lagrangianrelaxation and for Lagrangian decomposition
− Lagrangian decomposition: No gain in bound compared toLagrangian relaxation when capacity is relaxed
I [Letocart et al.(2012)]:The 0-1 bi-dimensional knapsack problem and the generalisedassignment problem
− Illustrates the concept− No full comparison of bounds, conclusions not possible
Our work this far:Find an application where we gain in strength compared to the strongestobtainable from Lagrangian relaxation and investigate further ...
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Related work
Very few papers on Lagrangian decomposition and column generation:
I [Pimentel et al.(2010)]:The multi-item capacitated lot sizing problem
− Branch-and-price implementations for two types of Lagrangianrelaxation and for Lagrangian decomposition
− Lagrangian decomposition: No gain in bound compared toLagrangian relaxation when capacity is relaxed
I [Letocart et al.(2012)]:The 0-1 bi-dimensional knapsack problem and the generalisedassignment problem
− Illustrates the concept− No full comparison of bounds, conclusions not possible
Our work this far:Find an application where we gain in strength compared to the strongestobtainable from Lagrangian relaxation and investigate further ...
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Fixed charge transportation problem (FCTP)
For each arc (i , j), i ∈ I , j ∈ J:uij= min(si , dj) = upper boundcij= unit cost for shippingfij= fixed cost for shipping
Variables:xij = amount shipped from source i to sink j , i ∈ I , j ∈ J
Concave cost function:
gij(xij) =
{fij + cijxij if xij > 00 if xij = 0
i ∈ I , j ∈ J
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Fixed charge transportation problem (FCTP)
For each arc (i , j), i ∈ I , j ∈ J:uij= min(si , dj) = upper boundcij= unit cost for shippingfij= fixed cost for shipping
Variables:xij = amount shipped from source i to sink j , i ∈ I , j ∈ J
Concave cost function:
gij(xij) =
{fij + cijxij if xij > 00 if xij = 0
i ∈ I , j ∈ J
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Fixed charge transportation problem (FCTP)
min∑i∈I
∑j∈J
gij(xij)
s.t.∑j∈J
xij = si i ∈ I
∑i∈I
xij = dj j ∈ J
xij ≥ 0 i ∈ I , j ∈ J
I Polytope of feasible solutions, minimisation of concave objective ⇒Optimal solution at an extreme point(can be non-global local optima at extreme points)
I MIP-formulation:A binary variable to indicate if there is flow on an arc or not
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
MIP-formulation of FCTP
min∑i∈I
∑j∈J
(cijxij + fijyij
)s.t.
∑j∈J
xij = si i ∈ I
∑i∈I
xij = dj j ∈ J
xij ≤ uijyij i ∈ I , j ∈ J
xij ≥ 0 i ∈ I , j ∈ J
yij ∈ {0, 1} i ∈ I , j ∈ J
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
The reformulation: variable splitting
Supply and demand side duplicates of the shipping variables: x sij and xdijIntroduce a parameter ν: 0 ≤ ν ≤ 1
min ν∑i∈I
∑j∈J
gij(xsij) + (1− ν)
∑j∈J
∑i∈I
gij(xdij )
s.t. x sij = xdij i ∈ I , j ∈ J∑j∈J
x sij = si i ∈ I
∑i∈I
xdij = dj j ∈ J
x sij , xdij ≥ 0 i ∈ I , j ∈ J,
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
The reformulation: inner representation
Let each column correspond to an extreme point of a set
X si = {x sij , j ∈ J |
∑j∈J x
sij = si , 0 ≤ x sij ≤ uij , j ∈ J}, i ∈ I ,
or of a set
X dj = {xdij , i ∈ I |
∑i∈I x
dij = dj , 0 ≤ xdij ≤ uij , i ∈ I}, j ∈ J
The flow from one source / to one sink is a convex combination ofextreme point flows, introduce:
λsip = convexity weight for extreme point p ∈ Psi of set X s
i , i ∈ I
and
λdjp = convexity weight for extreme point p ∈ Pdj of set X d
j , j ∈ J
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
The reformulation: column oriented formulation
min ν∑i∈I
∑j∈J
gij
∑p∈Ps
i
λsipxsijp
+ (1− ν)∑j∈J
∑i∈I
gij
∑p∈Pd
j
λdjpxdijp
s.t.
∑p∈Ps
i
x sijpλsip =
∑p∈Pd
j
xdijpλdjp i ∈ I , j ∈ J
∑p∈Ps
i
λsip = 1 i ∈ I
∑p∈Pd
j
λdjp = 1 j ∈ J
λsip ≥ 0 p ∈ Psi , i ∈ I
λdjp ≥ 0 p ∈ Pdj , j ∈ J
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
The reformulation: approximating the objective
The objective is bound below by its linearisation:
ν∑i∈I
∑j∈J
gij
∑p∈Ps
i
λsipxsijp
+ (1− ν)∑j∈J
∑i∈I
gij
∑p∈Pd
j
λdjpxdijp
≥ ν
∑i∈I
∑p∈Ps
i
∑j∈J
gij(xsijp)
λsip + (1− ν)∑j∈J
∑p∈Pd
j
∑i∈I
gij(xdijp)
λdjp
What is lost by the linearisation?
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
The reformulation: arc cost
True cost
In LP-relaxation of MIP-formulation
After our reformulation (here for supply side, similarly for sink side):I Extreme points: true cost
I Non-extreme points:
xsij =∑
p∈Psi
: λsip>0 x
sijpλ
sip
− if xsij > 0 and there is p ∈ Psi :
λsip > 0 and xsip = 0, then fij isdecreased by∆ =
∑p∈Ps
i: λs
ip>0, xsip=0 fijλsip
− otherwise: true cost
Example: For xsij = 4 = 6 23
+ 0 13
we have
xsij1 = 6, λsi1 = 23
; xsij2 = 0, λsi2 = 13
; ∆ = 13fij
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
The reformulation: arc cost
True cost In LP-relaxation of MIP-formulation
After our reformulation (here for supply side, similarly for sink side):I Extreme points: true cost
I Non-extreme points:
xsij =∑
p∈Psi
: λsip>0 x
sijpλ
sip
− if xsij > 0 and there is p ∈ Psi :
λsip > 0 and xsip = 0, then fij isdecreased by∆ =
∑p∈Ps
i: λs
ip>0, xsip=0 fijλsip
− otherwise: true cost
Example: For xsij = 4 = 6 23
+ 0 13
we have
xsij1 = 6, λsi1 = 23
; xsij2 = 0, λsi2 = 13
; ∆ = 13fij
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
The reformulation: arc cost
True cost In LP-relaxation of MIP-formulation
After our reformulation (here for supply side, similarly for sink side):I Extreme points: true cost
I Non-extreme points:
xsij =∑
p∈Psi
: λsip>0 x
sijpλ
sip
− if xsij > 0 and there is p ∈ Psi :
λsip > 0 and xsip = 0, then fij isdecreased by∆ =
∑p∈Ps
i: λs
ip>0, xsip=0 fijλsip
− otherwise: true cost
Example: For xsij = 4 = 6 23
+ 0 13
we have
xsij1 = 6, λsi1 = 23
; xsij2 = 0, λsi2 = 13
; ∆ = 13fij
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
The reformulation: arc cost
True cost In LP-relaxation of MIP-formulation
After our reformulation (here for supply side, similarly for sink side):I Extreme points: true cost
I Non-extreme points:
xsij =∑
p∈Psi
: λsip>0 x
sijpλ
sip
− if xsij > 0 and there is p ∈ Psi :
λsip > 0 and xsip = 0, then fij isdecreased by∆ =
∑p∈Ps
i: λs
ip>0, xsip=0 fijλsip
− otherwise: true cost
Example: For xsij = 4 = 6 23
+ 0 13
we have
xsij1 = 6, λsi1 = 23
; xsij2 = 0, λsi2 = 13
; ∆ = 13fij
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
The reformulation: column generation subproblem
For a single source i ∈ I (and similarly for the sinks):
min∑j∈J
(νgij(x
sij)− αijx
sij
)− βi
s.t.∑j∈J
x sij = si
x sij ≤ uij j ∈ J
x sij ≥ 0 j ∈ J
I Concave minimization problem with an optimal solution at anextreme point of X s
i
I MIP-formulation:Binary variable to indicate if there is flow on an arc or not
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Theoretical strength
[Guignard and Kim(1987)]: The Lagrangian decomposition bound(= convexification over source and sink side )as least as strong as the strongest of
I the Lagrangian relaxation bound when Ax ≤ b is relaxed(= convexification over source side )
I the Lagrangian relaxation bound when Cx ≤ d is relaxed(= convexification over sink side )
and there is a chance that it is stronger!
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Theoretical strength: type of strength?
The constraints in the MIP-formulation of the column generationsubproblem (source side, similarly for the sink side):
∑j∈J
x sij = si
x sij ≤ uijysij , j ∈ J
y sij ∈ {0, 1}, j ∈ J
0 ≤ x sij , j ∈ J
+
Implied inequality:∑j∈J
uijysij ≥ si =
∑j∈J
x sij = si
x sij ≤ uijysij , j ∈ J∑
j∈J
uijysij ≥ si
y sij ∈ {0, 1}, j ∈ J
0 ≤ x sij , j ∈ J
I Knapsack constraints over the binary variables
I At least (exactly?) that type of strength
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Theoretical strength: type of strength?
The constraints in the MIP-formulation of the column generationsubproblem (source side, similarly for the sink side):
∑j∈J
x sij = si
x sij ≤ uijysij , j ∈ J
y sij ∈ {0, 1}, j ∈ J
0 ≤ x sij , j ∈ J
+
Implied inequality:∑j∈J
uijysij ≥ si
=
∑j∈J
x sij = si
x sij ≤ uijysij , j ∈ J∑
j∈J
uijysij ≥ si
y sij ∈ {0, 1}, j ∈ J
0 ≤ x sij , j ∈ J
I Knapsack constraints over the binary variables
I At least (exactly?) that type of strength
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Theoretical strength: type of strength?
The constraints in the MIP-formulation of the column generationsubproblem (source side, similarly for the sink side):
∑j∈J
x sij = si
x sij ≤ uijysij , j ∈ J
y sij ∈ {0, 1}, j ∈ J
0 ≤ x sij , j ∈ J
+
Implied inequality:∑j∈J
uijysij ≥ si =
∑j∈J
x sij = si
x sij ≤ uijysij , j ∈ J∑
j∈J
uijysij ≥ si
y sij ∈ {0, 1}, j ∈ J
0 ≤ x sij , j ∈ J
I Knapsack constraints over the binary variables
I At least (exactly?) that type of strength
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Empirical strength
Comparison:
I Convexification over both sides = Lagrangian decomposition
I Convexifiation over one side, see bounds obtained by[Roberti et al.(2015)] (A new formulation based on extreme flowpatterns from each source; derive valid inequalities; exact branchand price; column generation to compute lower bounds)
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Empirical strength
Table: Instance characteristics
instance total r. v. c. r. f. c.type ID size supply inf sup inf sup
Roberti-set3 Table5-(1-10) 70×70 682-819 0 0 200 800Roberti-set3 Table6-(1-10) 70×70 705-760 7 32 200 800Roberti-set3 Table7-(1-10) 70×70 654-808 18 83 200 800
r.v.c. = range of variable costs; r.f.c. = range of fixed costs
I Compare LBDs by comparing gaps (relative deviations) calculated as(UBD-LBD)/LBD in percent
I The UBDs used are the best known, most of them are verified to beoptimal
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Empirical strengthinstance cost gap (%)
ID ratio (%) FCTP-LP Roberti et al. our
Table5-1 100 19.4 9.7 6.0Table5-2 100 16.7 10.6 5.2Table5-3 100 16.3 10.0 5.7Table5-4 100 18.9 8.8 4.5Table5-5 100 17.3 9.1 4.5Table5-6 100 18.5 9.1 4.2Table5-7 100 18.8 10.4 5.8Table5-8 100 17.2 8.0 4.4Table5-9 100 16.2 8.1 4.4Table5-10 100 16.8 9.7 5.4
AVG 17.6 9.3 5.0
instance cost gap (%)
ID ratio (%) FCTP-LP Roberti et al. our
Table6-1 79 16.1 10.3 5.0Table6-2 78 14.0 8.2 4.7Table6-3 78 16.6 8.3 4.8Table6-4 79 12.9 8.6 4.0Table6-5 79 16.4 8.4 4.7Table6-6 78 14.5 7.9 4.6Table6-7 78 15.6 6.8 4.8Table6-8 79 16.0 9.1 5.6Table6-9 79 15.5 9.7 4.4Table6-10 79 14.5 8.2 4.7
AVG 15.2 8.6 4.7
instance cost gap (%)
ID ratio (%) FCTP-LP Roberti et al. our
Table7-1 58 10.0 5.4 3.5Table7-2 58 12.9 8.1 6.2Table7-3 59 11.2 5.6 3.5Table7-4 60 13.3 7.9 4.8Table7-5 60 13.7 6.4 5.3Table7-6 58 10.1 6.4 3.5Table7-7 59 11.2 7.1 3.9Table7-8 59 11.5 6.6 3.9Table7-9 59 11.4 6.9 4.4Table7-10 59 10.8 6.9 3.1
AVG 11.6 6.7 4.2
Average improvement in gap thanks toconvexification,”from first side” + ”from second side”
I Table5: 47% + 46%
I Table6: 43% + 45%
I Table7: 42% + 37%
Both convexifications contribute!
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Other observations this far
I Property of the dual function: Constant along the direction e
I Stabilsation: No improvement – the opposite!
I Because of the property of the dual function: Tried regularisationinstead to favour solutions with a small l1-norm
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Conclusions ...
... this far:
I Strong formulation for the fixed charge transportation problem
I Empirically we gain significantly in strength from the convexificationover both sides
Further studies:
I Properties of the dual function, cf. experiences from subgradientmethods?
I Understand the effects of stabilization. Customized techniques?
I How strength depends on instance characteristics
Elina Ronnberg
Introduction and background The fixed charge transportation problem Concluding comments
Bibliography
Guignard M, Kim S (1987) Lagrangean decomposition: a model yielding stronger lagrangean bounds. Mathematical Programming
39 (2):215–228.
Letocart L, Nagih A, Touati-Moungla N (2012) Dantzig-Wolfe and Lagrangian decompositions in integer linear programming.
International Journal of Mathematics in Operational Research 4(3):247–262.
Pimentel CMO, Alvelos FP, de Carvalho JMV (2010) Comparing Dantzig-Wolfe decompositions and branch-and-price algorithms
for the multi-item capacitated lot sizing problem. Optimization Methods and Software 25(2):299–319.
Roberti R, Bartolini E, Mingozzi A (2015) The fixed charge transportation problem: an exact algorithm based on a new integer
programming formulation. Management Science, forthcoming.
Thanks for listening!
Elina Ronnberg